We are interested in vacuum solutions for massive scalar field inside a
Casimir apparatus in an axially symmetric stationary spacetime. Following
Sloane and Chandrasekhar [1]}, [2]}, we demand that such
spacetime is stationary, \(x^{0}\rightarrow x^{0}+c_{0}\) , has axial symmetry,
\(x^{1}\rightarrow x^{1}+c_{1}\)  and is invariant under simultaneous reflection
with respect to both \(x^{0}\)  and \(x^{1}\) , \(x^{0}\rightarrow -x^{0}\)  and
\(x^{1}\rightarrow -x^{1}\) . Under these assumptions, the most general line
element is given by:
\(ds^{2}=g_{00}\left( dx^{0}\right) ^{2}+2g_{01}dx^{0}dx^{1}+g_{11}\left(dx^{1}\right) ^{2}+g_{22}\left( dx^{2}\right) ^{2}+g_{33}\left(dx^{3}\right) ^{2}\,, \) 
